{"id":57,"date":"2023-09-17T07:24:22","date_gmt":"2023-09-17T07:24:22","guid":{"rendered":"https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/"},"modified":"2023-09-17T07:24:22","modified_gmt":"2023-09-17T07:24:22","slug":"exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/","title":{"rendered":"Como fatorar polin\u00f4mios (fatora\u00e7\u00e3o polinomial)"},"content":{"rendered":"<p>Nesta p\u00e1gina explicamos como fatorar qualquer tipo de polin\u00f4mio. Veremos primeiro como fatorar um polin\u00f4mio com a regra de Ruffini, depois passaremos a como s\u00e3o fatorados polin\u00f4mios sem termo independente, depois analisaremos as fatora\u00e7\u00f5es de polin\u00f4mios raiz com fra\u00e7\u00f5es e, por fim, os casos especiais de fatora\u00e7\u00f5es (not\u00e1veis identidades, fatora\u00e7\u00e3o por agrupamento, trin\u00f4mios, etc.). Todas as explica\u00e7\u00f5es s\u00e3o feitas com exemplos e, al\u00e9m disso, ao final voc\u00ea poder\u00e1 praticar com os exerc\u00edcios resolvidos passo a passo para fatorar polin\u00f4mios. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%C2%BFQue-es-la-factorizacion-de-polinomios\"><\/span> O que \u00e9 fatora\u00e7\u00e3o polinomial?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>A fatora\u00e7\u00e3o polinomial \u00e9 uma t\u00e9cnica usada em matem\u00e1tica para decompor um polin\u00f4mio no produto de fatores.<\/strong> <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/factorisation-de-polynomes-factorisation.png\" alt=\"fatora\u00e7\u00e3o de polin\u00f4mios (fatora\u00e7\u00e3o de polin\u00f4mios)\" class=\"wp-image-1177\" width=\"196\" height=\"197\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Fatorar polin\u00f4mios \u00e9 muito \u00fatil porque \u00e9 mais f\u00e1cil realizar opera\u00e7\u00f5es com polin\u00f4mios fatorados.<\/p>\n<p> Agora que sabemos o que \u00e9 fatora\u00e7\u00e3o polinomial, vamos ver como os polin\u00f4mios s\u00e3o fatorados. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Como-factorizar-polinomios-con-la-regla-de-Ruffini\"><\/span> Como fatorar polin\u00f4mios com a regra de Ruffini<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Obviamente, para entender como fatorar um polin\u00f4mio com a regra de Ruffini, primeiro voc\u00ea deve saber <a href=\"https:\/\/mathority.org\/pt\/regras-resolvidas-exemplos-exercicios-ruffini\/\"><strong><span style=\"text-decoration: underline;\">como aplicar a regra de Ruffini<\/span><\/strong><\/a> . Portanto, deixamos este link caso voc\u00ea queira primeiro revisar como era o procedimento.<\/p>\n<p> Para <strong>fatorar um polin\u00f4mio,<\/strong> as seguintes etapas devem ser seguidas:<\/p>\n<ol style=\"color:#ff5733; font-weight: bold;\">\n<li style=\"margin-bottom:18px\"> <span style=\"color:#000000;font-weight: normal;\">As ra\u00edzes do polin\u00f4mio s\u00e3o calculadas de acordo com a regra de Ruffini.<\/span><\/li>\n<li style=\"margin-bottom:18px\"> <span style=\"color:#000000;font-weight: normal;\">Cada raiz encontrada do tipo x=a \u00e9 expressa na forma de um fator (xa).<\/span><\/li>\n<li> <span style=\"color:#000000;font-weight: normal;\">O polin\u00f4mio fatorado \u00e9 o produto de todos os fatores encontrados multiplicado pelo coeficiente do termo de maior grau do polin\u00f4mio n\u00e3o ponderado.<\/span><\/li>\n<\/ol>\n<p> Para que voc\u00ea veja como isso \u00e9 feito e entenda melhor o procedimento de fatora\u00e7\u00e3o de polin\u00f4mios, abaixo voc\u00ea encontrar\u00e1 um exemplo concreto explicado passo a passo:<\/p>\n<ul>\n<li> Fatore o seguinte polin\u00f4mio: <\/li>\n<\/ul>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/polynome-non-factorise-2.jpg\" alt=\"polin\u00f4mio n\u00e3o ponderado\" class=\"wp-image-1221\" width=\"262\" height=\"31\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> A primeira coisa a fazer \u00e9 calcular as ra\u00edzes ou zeros do polin\u00f4mio. Para isso, precisamos encontrar os <strong>divisores do termo independente do polin\u00f4mio<\/strong> , que neste caso s\u00e3o \u00b11, \u00b12 e \u00b14. <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/factorisation-de-polynomes-pdf-2.jpg\" alt=\"fatora\u00e7\u00e3o de polin\u00f4mios pdf\" class=\"wp-image-1222\" width=\"383\" height=\"133\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Agora sabemos, gra\u00e7as ao teorema do resto e dos fatores, que se o resto da divis\u00e3o do polin\u00f4mio por um desses valores for igual a 0, isso significa que esse valor \u00e9 uma raiz do polin\u00f4mio.<\/p>\n<p class=\"has-background\" style=\"background-color:#ffebee\"> Devemos, portanto, dividir o polin\u00f4mio por cada um dos divisores do termo independente com a regra de Ruffini e ver em quais casos o resto \u00e9 zero.<\/p>\n<p> Por exemplo, come\u00e7amos aplicando a regra de Ruffini com <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0fc5c5d38d0edca51c40a0f9db16d55f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=+1:\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"65\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/factorisation-de-polynomes-avec-la-regle-de-ruffini.jpg\" alt=\"\" class=\"wp-image-1227\" width=\"315\" height=\"140\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Neste caso, o resto (ou res\u00edduo) da divis\u00e3o \u00e9 zero, ent\u00e3o<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3330a01aa4d7d81947b71297d8623d3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00c9 uma raiz do polin\u00f4mio. \u2705<\/p>\n<p> Perfeito, j\u00e1 temos uma raiz do polin\u00f4mio, s\u00f3 falta determinar as demais ra\u00edzes restantes. Para fazer isso, usamos a regra de Ruffini com outro divisor do termo independente, por exemplo<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6de4e73609a66312d9714a253f9ae3a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=-1.\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"61\" style=\"vertical-align: 0px;\"><\/p>\n<p> Al\u00e9m disso, n\u00e3o h\u00e1 necessidade de usar o m\u00e9todo de Ruffini com o polin\u00f4mio inteiro, mas podemos continuar de onde paramos no passo anterior: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/factorisation-des-polynomes-pas-a-pas.jpg\" alt=\"fatorar polin\u00f4mios passo a passo\" class=\"wp-image-1225\" width=\"259\" height=\"221\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Por\u00e9m, neste caso, ao dividir por<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ad143a0d979362a51b48a48c9ca9f59e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=-1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<p> o resto obtido \u00e9 diferente de 0, ent\u00e3o<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ad143a0d979362a51b48a48c9ca9f59e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=-1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<p> N\u00e3o \u00e9 uma raiz do polin\u00f4mio. \u274c<\/p>\n<p> Devemos portanto tentar outro valor, por exemplo fazemos a regra de Ruffini com <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-feebbb5a0ef01c12d307ae7005579405_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=+2:\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"65\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/factoriser-les-polynomes-pas-a-pas.jpg\" alt=\"fatorar polin\u00f4mios passo a passo\" class=\"wp-image-1232\" width=\"219\" height=\"217\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Neste caso, novamente obtemos um resto zero, ent\u00e3o<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c657687cbbf5ea9a7545edb42190e592_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00c9 tamb\u00e9m uma raiz do polin\u00f4mio.<\/p>\n<p> E continuamos a aplicar o mesmo procedimento. Agora verificamos se<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-01f282abd343bbe6b83c45e54b86c6ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=-2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 uma raiz do polin\u00f4mio ou n\u00e3o: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exemple-factorisation-de-polynomes.jpg\" alt=\"exemplo de fatora\u00e7\u00e3o de polin\u00f4mios\" class=\"wp-image-1250\" width=\"219\" height=\"303\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Ao dividir por<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-01f282abd343bbe6b83c45e54b86c6ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=-2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<p> Com a regra de Ruffini obtemos resto zero, ent\u00e3o<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-01f282abd343bbe6b83c45e54b86c6ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=-2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 uma raiz ou zero do polin\u00f4mio.<\/p>\n<p> N\u00e3o podemos, portanto, continuar a aplicar a regra de Ruffini, pois j\u00e1 encontramos todas as ra\u00edzes do polin\u00f4mio, que s\u00e3o: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/racines-polynome-factorise.jpg\" alt=\"ra\u00edzes de um polin\u00f4mio fatorado\" class=\"wp-image-1238\" width=\"352\" height=\"41\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Depois de determinarmos todas as ra\u00edzes do polin\u00f4mio, podemos fator\u00e1-lo. Para fazer isso, basta expressar cada raiz<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2b24e8b3f28f048c85d6ea0f32d59fff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"43\" style=\"vertical-align: 0px;\"><\/p>\n<p> na forma de um fator do tipo<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f629cb3501652d3b8e4d6a30d92b5d4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x-a)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"53\" style=\"vertical-align: -5px;\"><\/p>\n<p> , ou seja, para cada raiz voc\u00ea deve colocar um par\u00eantese com um<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> e a raiz mudou de sinal: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/comment-factoriser-un-polynome.jpg\" alt=\"como fatorar um polin\u00f4mio\" class=\"wp-image-1235\" width=\"416\" height=\"172\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> E agora que temos todas as ra\u00edzes expressas como fatores, devemos multiplicar todos os par\u00eanteses pelo coeficiente do termo de maior grau do polin\u00f4mio original: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/facteur-polynomes-coefficient-plus-grand-degre.jpg\" alt=\"fator polin\u00f4mios coeficiente de grau mais alto\" class=\"wp-image-1241\" width=\"365\" height=\"124\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Embora neste caso o coeficiente seja 1 e, portanto, n\u00e3o afete o resultado, \u00e9 importante lembrar de fazer esta multiplica\u00e7\u00e3o. Pois se o referido coeficiente fosse diferente de 1, o polin\u00f4mio fatorado mudaria e, portanto, ao n\u00e3o inserir o n\u00famero cometer\u00edamos um erro na fatora\u00e7\u00e3o do polin\u00f4mio.<\/p>\n<p> Resumindo, o polin\u00f4mio fatorado \u00e9: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/polymial-factorise-etape-par-etape-en-ligne.jpg\" alt=\"polin\u00f4mio fatorado passo a passo on-line\" class=\"wp-image-1242\" width=\"339\" height=\"39\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Factorizar-polinomios-sin-termino-independiente\"><\/span> Fatora\u00e7\u00e3o de polin\u00f4mios sem termo independente<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Acabamos de ver que o termo independente \u00e9 importante para a fatora\u00e7\u00e3o de polin\u00f4mios, pois permite identificar as poss\u00edveis ra\u00edzes do polin\u00f4mio. No entanto, como fatorar um polin\u00f4mio que n\u00e3o possui termo independente?<\/p>\n<p class=\"has-background\" style=\"background-color:#ffebee\"> Para <strong>fatorar um polin\u00f4mio sem termo independente<\/strong> , deve-se primeiro extrair o fator comum do polin\u00f4mio, depois extrair as ra\u00edzes do polin\u00f4mio sem o fator comum usando a regra de Ruffini.<\/p>\n<p> Escrito assim, pode parecer um pouco complicado, ent\u00e3o vamos resolver um exemplo passo a passo para que voc\u00ea veja como fatorar um polin\u00f4mio com um fator comum:<\/p>\n<ul>\n<li> Execute a decomposi\u00e7\u00e3o fatorial do seguinte polin\u00f4mio:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d9f7b2c27b1431f9362ee4268f48698e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) = x^4-3x^3-x^2+3x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"208\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Como voc\u00ea pode ver, o polin\u00f4mio do problema n\u00e3o possui um termo independente, ent\u00e3o temos que pegar o fator comum do polin\u00f4mio. Se olharmos atentamente, todos os elementos do polin\u00f4mio t\u00eam pelo menos um<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-038741496726a75b03e91a2e030b0287_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x,\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -4px;\"><\/p>\n<p> ent\u00e3o o fator comum \u00e9<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a9cc293b28f198c32e0356b52e2e23bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> Portanto, ao extrair o fator comum do polin\u00f4mio, obtemos a seguinte express\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2c92203e4f8834fe75ccb4a71340ff7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) = x\\left(x^3-3x^2-x+3\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"217\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p> E uma vez extra\u00eddo o fator comum do polin\u00f4mio, aplicamos a regra de Ruffini para calcular as ra\u00edzes do polin\u00f4mio agrupadas entre par\u00eanteses (com o procedimento que vimos na se\u00e7\u00e3o anterior): <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/factoriser-les-polynomes-sans-terme-independant.jpg\" alt=\"polin\u00f4mios fatoriais sem termo independente\" class=\"wp-image-1251\" width=\"219\" height=\"303\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Portanto, as ra\u00edzes ou zeros do polin\u00f4mio entre par\u00eanteses s\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3146b4cef2e32a512e054760ad4fd3a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{x=+1 \\qquad x=-1 \\qquad x=+3}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"241\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> Portanto, para fatorar o polin\u00f4mio, basta substituir o polin\u00f4mio entre par\u00eanteses por suas ra\u00edzes na forma fatorial (conforme explicado na se\u00e7\u00e3o acima):<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-470b8a931d73b852bee700a6488af525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}P(x) = x\\left(x^3-3x^2-x+3\\right) \\\\[2ex]\\color{red} \\bm{\\downarrow} \\\\[2ex] \\bm{P(x) = x(x-1)(x+1)(x-3)}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"95\" width=\"234\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> E desta forma j\u00e1 fatoramos o polin\u00f4mio que n\u00e3o tinha termo de grau 0. Observe que a \u00fanica diferen\u00e7a \u00e9 que primeiro temos que extrair um fator comum, mas todos os passos seguintes s\u00e3o exatamente iguais.<\/p>\n<p> Por outro lado, voc\u00ea deve saber que<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00c9 tamb\u00e9m uma raiz do polin\u00f4mio, pois quando extra\u00edmos o fator comum, implica que uma das ra\u00edzes do polin\u00f4mio \u00e9<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d6889ee3f02f0af137641306363d2da7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=0.\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\"><\/p>\n<p> Portanto, todas as ra\u00edzes do polin\u00f4mio s\u00e3o as seguintes:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-20064fa13be3ed4c61c9b8e4cc4afe1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{x= 0 \\qquad x=+1 \\qquad x=-1 \\qquad x=+3}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"319\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> Na verdade, o polin\u00f4mio deve ter tantas ra\u00edzes quanto o seu grau indicar. Neste caso o polin\u00f4mio \u00e9 de grau 4 e portanto possui 4 ra\u00edzes. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Factorizar-polinomios-con-raices-racionales\"><\/span> Fatora\u00e7\u00e3o de polin\u00f4mios com ra\u00edzes racionais<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> At\u00e9 aqui vimos exemplos de fatora\u00e7\u00e3o de polin\u00f4mios com ra\u00edzes inteiras, por\u00e9m, um polin\u00f4mio tamb\u00e9m pode ter ra\u00edzes racionais, ou seja, com fra\u00e7\u00f5es. Vamos ver como esse tipo de fatora\u00e7\u00e3o polinomial \u00e9 resolvido com um exemplo:<\/p>\n<ul>\n<li> Fatore o seguinte polin\u00f4mio incompleto:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1fe7ec3cf4891c96dd472a3328c6a946_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) = 4x^3-7x+3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"159\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Como sempre, utilizamos a regra de Ruffini com os divisores do termo independente para tentar determinar as ra\u00edzes do polin\u00f4mio: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/factoriser-des-polynomes-avec-des-racines-rationnelles.jpg\" alt=\"Fatora\u00e7\u00e3o de polin\u00f4mios com ra\u00edzes racionais\" class=\"wp-image-1268\" width=\"226\" height=\"136\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Mas n\u00e3o podemos calcular mais ra\u00edzes com Ruffini, porque se tentarmos fazer Ruffini com todos os outros n\u00fameros divisores do termo independente obteremos um resto diferente de zero.<\/p>\n<p> Encontramo-nos, portanto, numa situa\u00e7\u00e3o em que s\u00f3 com<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3330a01aa4d7d81947b71297d8623d3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<p> o restante da divis\u00e3o equivale a 0, isso significa que o polin\u00f4mio pode ter ra\u00edzes fracion\u00e1rias. Para determinar essas ra\u00edzes poder\u00edamos aplicar Ruffini com fra\u00e7\u00f5es, por\u00e9m \u00e9 muito f\u00e1cil cometer erros nos c\u00e1lculos e \u00e9 por isso que nestes casos costumamos fazer o seguinte:<\/p>\n<p class=\"has-background\" style=\"background-color:#ffebee\"> Quando n\u00e3o podemos continuar aplicando a regra de Ruffini com ra\u00edzes inteiras, devemos igualar o \u00faltimo polin\u00f4mio obtido a 0 e resolver a equa\u00e7\u00e3o resultante. Portanto, as ra\u00edzes do polin\u00f4mio ser\u00e3o os valores encontrados na equa\u00e7\u00e3o.<\/p>\n<p> Por outro lado, se a equa\u00e7\u00e3o n\u00e3o tiver solu\u00e7\u00e3o, isso significa que o polin\u00f4mio n\u00e3o tem mais ra\u00edzes e, portanto, n\u00e3o pode ser totalmente fatorado. <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/factoriser-polynomes-equation.jpg\" alt=\"polin\u00f4mios fatoriais on-line\" class=\"wp-image-1269\" width=\"226\" height=\"215\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Portanto, definimos o polin\u00f4mio quociente igual a zero:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-86de558e97d35772c24d155a06770271_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4x^2+4x-3=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"131\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> E usamos a f\u00f3rmula da equa\u00e7\u00e3o quadr\u00e1tica para resolver a equa\u00e7\u00e3o resultante: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-982cd2e82d8511ceb1f93648c3ee61df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= \\cfrac{-b \\pm \\sqrt{b^2-4ac}}{2a}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"165\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c0dbd8f544bd58121914b28752b950d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= \\cfrac{-4 \\pm \\sqrt{4^2-4\\cdot 4\\cdot (-3)}}{2\\cdot 4}= \\cfrac{-4\\pm \\sqrt{16+48}}{8} = \\cfrac{-4 \\pm\\sqrt{64}}{8}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"484\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f52601e0daafdb92974cfbfe6613733b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle x = \\cfrac{-4 \\pm 8}{8} = \\begin{cases}  \\cfrac{-4+8}{8} = \\cfrac{4}{8} = \\cfrac{1}{2} \\\\[4ex]\\cfrac{-4-8}{8} = \\cfrac{-12}{8} = -\\cfrac{3}{2} \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"312\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> As ra\u00edzes do polin\u00f4mio s\u00e3o, portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-951f2eb0c362c411dbb364b814cd05a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=+1 \\qquad x=\\cfrac{1}{2} \\qquad x=-\\cfrac{3}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"230\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> O polin\u00f4mio, portanto, tem ra\u00edzes na forma de fra\u00e7\u00f5es.<\/p>\n<p> E uma vez que conhecemos todas as ra\u00edzes do polin\u00f4mio, podemos facilmente encontrar o polin\u00f4mio fatorado expressando cada raiz<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2b24e8b3f28f048c85d6ea0f32d59fff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"43\" style=\"vertical-align: 0px;\"><\/p>\n<p> na forma de um fator do tipo<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f629cb3501652d3b8e4d6a30d92b5d4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x-a)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"53\" style=\"vertical-align: -5px;\"><\/p>\n<p> , ou seja, para cada raiz voc\u00ea deve colocar um par\u00eantese com um<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> e a raiz mudou de sinal:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-47682def6b604adb2c1ec62c59181f05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle P(x)= 4\\left(x-1\\right)\\left(x-\\frac{1}{2}\\right)\\left(x+\\frac{3}{2}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"272\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Lembre-se que para fatorar um polin\u00f4mio, voc\u00ea tamb\u00e9m deve multiplicar seus fatores pelo coeficiente do termo de maior grau do polin\u00f4mio n\u00e3o fatorado, que neste caso \u00e9 4. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Casos-especiales-de-la-factorizacion-de-polinomios\"><\/span> Casos especiais de fatora\u00e7\u00e3o de polin\u00f4mios<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Normalmente, a regra de Ruffini (ou divis\u00e3o sint\u00e9tica) \u00e9 usada para fatorar um polin\u00f4mio, conforme explicado acima. Mas dependendo do polin\u00f4mio do problema, \u00e0s vezes voc\u00ea pode fazer a fatora\u00e7\u00e3o polinomial mais rapidamente. Veremos cada um desses casos espec\u00edficos a seguir. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Factorizacion-de-identidades-notables\"><\/span> Fatorando identidades not\u00e1veis<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Se observarmos que um polin\u00f4mio corresponde a uma identidade not\u00e1vel (ou a um produto not\u00e1vel) \u00e9 muito simples fator\u00e1-lo. Por\u00e9m, para poder fazer isso voc\u00ea deve dominar as <a href=\"https:\/\/mathority.org\/pt\/identidades-produtos-igualdades-notaveis-exercicios-resolvidos\/\"><strong><span style=\"text-decoration: underline;\">f\u00f3rmulas para identidades not\u00e1veis<\/span><\/strong><\/a> , caso contr\u00e1rio recomendo que voc\u00ea d\u00ea uma olhada neste link onde voc\u00ea n\u00e3o s\u00f3 encontrar\u00e1 as f\u00f3rmulas, mas tamb\u00e9m poder\u00e1 ver exemplos de not\u00e1veis. identidades e voc\u00ea ainda pode praticar exerc\u00edcios com elas resolvidas passo a passo.<\/p>\n<h4 class=\"wp-block-heading\"> Diferen\u00e7a de quadrados<\/h4>\n<p> Como voc\u00ea bem sabe, a f\u00f3rmula para a identidade not\u00e1vel da diferen\u00e7a de quadrados \u00e9 a seguinte:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0f923a8db837402f512b6289f9c55b22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a^2-b^2=(a+b)\\cdot (a-b)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"195\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Portanto, se encontrarmos um polin\u00f4mio que satisfa\u00e7a a express\u00e3o<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3e8c1ff5ed178c14d02192ff8c85b93b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a^2-b^2\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"54\" style=\"vertical-align: 0px;\"><\/p>\n<p> podem ser levados em considera\u00e7\u00e3o diretamente.<\/p>\n<p> Veja o seguinte exemplo em que uma diferen\u00e7a de quadrados \u00e9 levada em considera\u00e7\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ab016e5ab7f26bfdba5420de9eae026b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-9 = (x+3)(x-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"180\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Por outro lado, as ra\u00edzes do polin\u00f4mio s\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e067558315750d60883807860c2a2b63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=-3 \\qquad x=+3\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"149\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> Outros exemplos de fatora\u00e7\u00e3o de bin\u00f4mios que s\u00e3o diferen\u00e7as de quadrados: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2a390b06ab6932ccc4f5f3fbe7abdfd1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-4=(x+2)(x-2) \\quad \\longrightarrow \\quad \\text{ra\\'ices: } x=-2, +2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"397\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a8afd88f2265f4622a54d266f0a22e39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-16=(x+4)(x-4) \\quad \\longrightarrow \\quad \\text{ra\\'ices: } x=-4, +4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"407\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1288583c4a3c4ae7379771e36a25675a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-25=(x+5)(x-5) \\quad \\longrightarrow \\quad \\text{ra\\'ices: } x=-5, +5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"406\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h4 class=\"wp-block-heading\"> Quadrado de adi\u00e7\u00e3o e subtra\u00e7\u00e3o<\/h4>\n<p> Voc\u00ea j\u00e1 deve conhecer as f\u00f3rmulas para as duas principais identidades not\u00e1veis restantes: o quadrado da adi\u00e7\u00e3o e o quadrado da subtra\u00e7\u00e3o. <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-15\">\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"> <strong>Soma Quadrada<\/strong> <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9fa3b0c36dfc418214a76610055f0f6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a^2+2ab+b^2 = (a+b)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"185\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"> <strong>Quadrado de subtra\u00e7\u00e3o<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-64a8720170a57c4dadc65bb16a53a40f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a^2-2ab+b^2= (a-b)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"185\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p> Assim, se percebermos que um polin\u00f3mio corresponde a uma destas duas identidades not\u00e1veis, podemos fator\u00e1-lo diretamente. Veja os seguintes exemplos: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-18\">\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-68811281a183bb1881b2fa5a799f4c86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2+6x+9 = (x+3)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"174\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> Raiz dupla: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7e135cd6350a4c21195c621240f7aee7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=-3\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-835d8f2fbf4c3d39b940c19563819e62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-8x+16 = (x-4)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"183\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> Raiz dupla:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2145acc2878ed61214887e120f2485b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=4\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p> Identificar esses tipos de produtos not\u00e1veis \u00e9 um pouco mais dif\u00edcil. Um truque \u00e9 verificar se o termo independente do polin\u00f4mio \u00e9 o quadrado de algum n\u00famero e se o termo de grau superior \u00e9 o quadrado de um mon\u00f4mio (geralmente<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> ), neste caso basta verificar que \u00e9 verdade que<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ddef4007f116e84febe922aa24a12bca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2ab\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 igual ao final do diploma intermedi\u00e1rio.<\/p>\n<p> Por exemplo, se tivermos o seguinte polin\u00f4mio:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9a4bd9309bbef0daf8a78bdf68d3dda5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2+10x+25\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"106\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> Neste caso, s\u00f3 pode ser o quadrado de uma soma, pois todos os elementos do polin\u00f4mio s\u00e3o positivos. Ent\u00e3o a vari\u00e1vel<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> da f\u00f3rmula deve ser 5, pois \u00e9 a raiz do termo independente, e a vari\u00e1vel<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> deve ser<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> , j\u00e1 que \u00e9 a raiz do termo pode grau.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-af4b0855281c4c0bc0f6044b1b3c33b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a=\\sqrt{x^2} = x\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -1px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3de5e4f309976794a981c07b460b7ced_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b=\\sqrt{25} = 5\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"95\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> Tudo o que temos que fazer agora \u00e9 provar que a f\u00f3rmula do quadrado da soma se cumpre com o termo de grau intermedi\u00e1rio:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6bc856a1d19034326a8cbf497ccf1a70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2ab = 10x \\ \\color{blue} \\bm{?}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"123\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5aebd0e6397d5edb14bf61460fe20884_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2ab = 2\\cdot x \\cdot 5 = 10 x\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"155\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u2705<\/p>\n<p> A f\u00f3rmula para o produto not\u00e1vel \u00e9 satisfeita, ent\u00e3o o polin\u00f4mio fatorado \u00e9:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-be9ab911bb8121aa0797b837f789fbc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2+10x+25 = (x+5)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"192\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> E a raiz deste polin\u00f4mio \u00e9<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f0fdc1717d47916064f25e11eb18b433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=-5,\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"61\" style=\"vertical-align: -4px;\"><\/p>\n<p> que \u00e9 uma raiz dupla porque seu fator \u00e9 elevado ao quadrado (\u00e9 repetido duas vezes).<\/p>\n<p> Abaixo est\u00e3o mais exemplos de fatora\u00e7\u00e3o de trin\u00f4mios quadrados perfeitos: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8f3df048ecccff6794ae04aebd3098b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-4x+4=(x-2)^2 \\quad \\longrightarrow \\quad \\text{ra\\'iz doble: } x=+2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"393\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e42fc53bf079ddd1f94f39f1d3cbb79e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2+14x+49=(x+7)^2 \\quad \\longrightarrow \\quad \\text{ra\\'iz doble: } x=-7\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"412\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36668f39fcb48e7113e0c9502dd1e98e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"9x^2-12x+4=(3x-2)^2 \\quad \\longrightarrow \\quad \\text{ra\\'iz doble: } x=+\\cfrac{2}{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"422\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Factorizacion-de-trinomios-de-segundo-grado\"><\/span> Fatora\u00e7\u00e3o de trin\u00f4mios de segundo grau<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Como acabamos de ver, \u00e0s vezes existem trin\u00f4mios que s\u00e3o quadrados perfeitos e podem ser fatorados diretamente com as f\u00f3rmulas para identidades not\u00e1veis. Mas a maioria dos trin\u00f4mios n\u00e3o s\u00e3o produtos not\u00e1veis, ent\u00e3o como fatoramos esses casos de polin\u00f4mios?<\/p>\n<p class=\"has-background\" style=\"background-color:#ffebee\"> Para fatorar um polin\u00f4mio quadr\u00e1tico n\u00e3o \u00e9 necess\u00e1rio aplicar o m\u00e9todo de Ruffini, basta igualar o polin\u00f4mio a zero e resolver a equa\u00e7\u00e3o quadr\u00e1tica resultante. As solu\u00e7\u00f5es da equa\u00e7\u00e3o ser\u00e3o, portanto, as ra\u00edzes do polin\u00f4mio.<\/p>\n<p> Por exemplo, se formos solicitados a fatorar o seguinte polin\u00f4mio de grau 2:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36b88a11b04e820a1154acc759f76526_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) = x^2+2x-15\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"159\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Em vez de usar Ruffini, definimos o polin\u00f4mio igual a 0:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-adffe34777fbc2f8972feba6d1978069_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2+2x-15=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"131\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> E agora usamos a f\u00f3rmula da equa\u00e7\u00e3o do 2\u00ba grau para encontrar as solu\u00e7\u00f5es da equa\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-982cd2e82d8511ceb1f93648c3ee61df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= \\cfrac{-b \\pm \\sqrt{b^2-4ac}}{2a}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"165\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b547af034f4845272c3029db9ac44655_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= \\cfrac{-2 \\pm \\sqrt{2^2-4\\cdot 1\\cdot (-15)}}{2\\cdot 1}= \\cfrac{-2\\pm \\sqrt{4+60}}{2} = \\cfrac{-2 \\pm\\sqrt{64}}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"484\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cd949c11577e283ded1f45e1ba2fa35b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle x = \\cfrac{-2 \\pm 8}{2} = \\begin{cases}  \\cfrac{-2+8}{2} = \\cfrac{6}{2} = 3 \\\\[4ex]\\cfrac{-2-8}{2} = \\cfrac{-10}{2} = -5 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"310\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> As ra\u00edzes do polin\u00f4mio s\u00e3o, portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1b4a33e3391cf08034f5eb56db357d48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=3 \\qquad x=-5\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"134\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> E finalmente, a fatora\u00e7\u00e3o polinomial \u00e9: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0edc3070d203b07e2a6733ea20d32e4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) =(x-3)(x+5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"170\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Factorizacion-de-trinomios-de-cuarto-grado-con-exponentes-pares\"><\/span> Fatora\u00e7\u00e3o de trin\u00f4mios de quarto grau com expoentes pares<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Como no caso anterior, para fatorar um polin\u00f4mio de quarto grau com expoentes pares, precisamos igualar o polin\u00f4mio a zero e resolver a equa\u00e7\u00e3o biquadrada. Para que os valores encontrados correspondam \u00e0s ra\u00edzes do polin\u00f4mio.<\/p>\n<p> Como exemplo, fatoraremos o seguinte polin\u00f4mio de grau 4:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-12410a6427e804a3da27995f9f1db53b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) =x^4-5x^2+4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"158\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Primeiro, definimos o polin\u00f4mio igual a zero:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-538c182becda938701ff5b1d4b18cfe8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^4-5x^2+4=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"129\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> Agora precisamos resolver a equa\u00e7\u00e3o biquadrada. Para fazer isso, fazemos uma altera\u00e7\u00e3o de vari\u00e1vel:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2b3e2aab5c304c6d5ff967115a60f95f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2=t\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"47\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-94dcc4d964c45f24e57e4d079f7fc1e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"t^2-5t+4=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"114\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> Resolvemos a equa\u00e7\u00e3o quadr\u00e1tica com a f\u00f3rmula: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-85e35ac03383600ab404fc9893a559e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"t= \\cfrac{-b \\pm \\sqrt{b^2-4ac}}{2a}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"161\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-77cd61b54d5510fe9247d465ffea7ff5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"t= \\cfrac{-(-5) \\pm \\sqrt{(-5)^2-4\\cdot 1\\cdot 4}}{2\\cdot 1}= \\cfrac{5\\pm \\sqrt{25-16}}{2} = \\cfrac{5 \\pm\\sqrt{9}}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"456\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5d5dada92a4b578d23d0e32ab6dac388_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle t = \\cfrac{5 \\pm 3}{2} = \\begin{cases}  \\cfrac{5+3}{2} = \\cfrac{8}{2} = 4 \\\\[4ex]\\cfrac{5-3}{2} = \\cfrac{2}{2} = 1 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"219\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Cancelamos a mudan\u00e7a de vari\u00e1vel para calcular as ra\u00edzes: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2b3e2aab5c304c6d5ff967115a60f95f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2=t\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"47\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-21\">\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c269e23a1070b3e5556abece040af75a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2=4\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3cf3da03bf703f0090af0eeb3709440f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\sqrt{4}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"58\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c06a55e3acdd1e283973786926b27716_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\pm 2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-959000af33497314f9a59a9bed2a19c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2=1\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"49\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-32986b6409a97918295bdd495b6cb869_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\sqrt{1}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"58\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b2e5d1349000e44cc1988f98254e0389_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\pm 1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p> As ra\u00edzes do polin\u00f4mio s\u00e3o, portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cf159b0d3425624012695374c1a90482_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=+2 \\qquad x=-2 \\qquad x=+1 \\qquad x=-1\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"331\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> E uma vez que conhecemos as ra\u00edzes ou zeros do polin\u00f4mio, n\u00f3s o fatoramos expressando suas ra\u00edzes algebricamente na forma de fatores: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7eaaf4ac3b4c848127ddfa0ab9d978c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) =(x-1)(x+1)(x-2)(x+2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"278\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Factorizacion-de-polinomios-por-agrupacion\"><\/span> Fatora\u00e7\u00e3o de polin\u00f4mios por agrupamento<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Em alguns casos muito especiais, uma f\u00f3rmula pode ser usada para fatorar um tipo muito particular de polin\u00f4mio.<\/p>\n<p> Se tivermos um polin\u00f4mio da seguinte forma:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0760e62da77badd13476ae11abad85a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-ax- bx+ab\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"137\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> Podemos simplificar o polin\u00f4mio removendo o fator comum:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d2913aa00fe4914d11171a6d74a0f239_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-ax- bx+ab = x(x-a)-b(x-a)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"309\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> E o polin\u00f4mio pode ser ainda mais simplificado extraindo o fator comum uma segunda vez:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d9e72259656373174da6552b009fad25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x(x-a)-b(x-a) =(x-a)\\cdot (x-b)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"293\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Desta forma conseguimos fatorar o polin\u00f4mio sem aplicar Ruffini ou qualquer outro m\u00e9todo. E as ra\u00edzes desse polin\u00f4mio seriam:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3ca29d3d668899f2ab265da4648a569b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=a \\qquad x=b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"120\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Agora vamos ver este m\u00e9todo com um exemplo num\u00e9rico:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3821ceccc53eb607d29f39c944625a75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-3x-2x+6\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"130\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> Primeiro, removemos o fator comum com<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> e com 2:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-13401ca14699ffd34366db3cfbf2aba8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-3x-2x+6 = x(x-3)-2(x-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"302\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> E como agora<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b531622467ab2607de193e88e4c52463_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"53\" style=\"vertical-align: -5px;\"><\/p>\n<p> \u00e9 um fator comum do polin\u00f4mio, extra\u00edmos o fator comum de<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d50ab8dee5078155995ce61e884141ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x-3):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"62\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7cfb180b19b0bf7dfb17d76d6869ab26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x(x-3)-2(x-3)=(x-3)\\cdot (x-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"294\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> As ra\u00edzes do polin\u00f4mio s\u00e3o, portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f70b397f33c52e3813f89b96ce0ae44c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=3 \\qquad x=2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"120\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Este m\u00e9todo tamb\u00e9m \u00e9 chamado de fatora\u00e7\u00e3o de polin\u00f4mios por extra\u00e7\u00e3o de fator comum duplo. Embora este seja um procedimento muito r\u00e1pido, n\u00e3o recomendamos realizar este tipo de fatora\u00e7\u00e3o porque erros s\u00e3o frequentemente relatados ao fatorar com este m\u00e9todo. Al\u00e9m disso, como vimos acima, um polin\u00f4mio de grau 2 tamb\u00e9m pode ser fatorado resolvendo uma equa\u00e7\u00e3o quadr\u00e1tica simples. Resumindo, nada acontece se voc\u00ea n\u00e3o entender bem esse m\u00e9todo.<\/p>\n<p> Por fim, deve-se notar que existem ainda outros m\u00e9todos de fatora\u00e7\u00e3o polinomial mais complexos, como o algoritmo LLL, o m\u00e9todo Kronecker e o m\u00e9todo Trager, que n\u00e3o s\u00e3o explicados aqui devido \u00e0 sua dificuldade matem\u00e1tica. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejercicios-resueltos-de-factorizacion-de-polinomios\"><\/span> Exerc\u00edcios resolvidos sobre fatora\u00e7\u00e3o de polin\u00f4mios<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Depois de ver todos os tipos de fatora\u00e7\u00e3o de polin\u00f4mios, recomendamos que voc\u00ea pratique a tentativa de resolu\u00e7\u00e3o de exerc\u00edcios. \u00c9 por isso que preparamos abaixo v\u00e1rios exerc\u00edcios resolvidos passo a passo para fatora\u00e7\u00e3o de polin\u00f4mios. Lembre-se que se voc\u00ea tiver alguma d\u00favida, pode escrev\u00ea-la nos coment\u00e1rios e responderemos rapidamente.<\/p>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Fa\u00e7a a fatora\u00e7\u00e3o do seguinte polin\u00f4mio de grau 3: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-797de8956ba7ef9e806e044d8969d8eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=x^3-3x^2-6x+8\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"199\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__DDF5FF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#DDF5FF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> \u00c9 um polin\u00f4mio completo, ordenado, de terceiro grau e, em \u00faltima an\u00e1lise, independente. Portanto, aplicamos o m\u00e9todo de Ruffini para determinar as ra\u00edzes do polin\u00f4mio: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercices-resolus-pas-a-pas-de-factorisation-de-polynomes.jpg\" alt=\"exerc\u00edcios passo a passo para fatorar polin\u00f4mios\" class=\"wp-image-1321\" width=\"218\" height=\"304\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> As ra\u00edzes do polin\u00f4mio s\u00e3o, portanto, as seguintes:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d3c985adaa6800a87a49d517c55c3bc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=+1 \\qquad x=-2 \\qquad x=+4\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"241\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A fatora\u00e7\u00e3o polinomial \u00e9, portanto: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3b7e45b1e7ee51b6a0d44615479dcaac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)= 1 \\cdot (x-1)\\cdot (x+2) \\cdot (x-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"271\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-198255688121f7bf7bc89a1e887b22ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{P(x)= (x-1)(x+2)(x-4)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"224\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 2<\/h3>\n<p> Calcule a fatora\u00e7\u00e3o do seguinte polin\u00f4mio de grau 4: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9f752dc1ebbc1db3f61b070a2928503b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=x^4+x^3-7x^2-x+6\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"230\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__DDF5FF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#DDF5FF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> \u00c9 um polin\u00f4mio de quarto grau e com termo independente, portanto utilizamos o m\u00e9todo de Ruffini para encontrar as ra\u00edzes do polin\u00f4mio: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercices-de-factorisation-polynomiale-pdf.jpg\" alt=\"exerc\u00edcios de fatora\u00e7\u00e3o polinomial pdf\" class=\"wp-image-1324\" width=\"251\" height=\"382\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> As ra\u00edzes do polin\u00f4mio s\u00e3o, portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-315cfedf34708088883c3373931e31ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=+1 \\qquad x=-1 \\qquad x=2 \\qquad x=-3\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"319\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E quando fatoramos o polin\u00f4mio, ficamos com: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5054d766aff29b0f07e5af1fbab6c704_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)= 1 \\cdot (x-1)\\cdot (x+1)\\cdot (x-2) \\cdot (x+3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"339\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e56ee9448e04783abf4fdd7e79b555f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{P(x)= (x-1)(x+1)(x-2)(x+3)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"278\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 3<\/h3>\n<p> Encontre a fatora\u00e7\u00e3o do seguinte polin\u00f4mio de quarto grau: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a5e2ea1abb94e9122f4a71f70fda9c42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=x^4-2x^3-13x^2-10x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"234\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__DDF5FF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#DDF5FF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Neste caso o polin\u00f4mio n\u00e3o possui termo independente, devemos primeiro extrair um fator comum:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-44f0fffdcd54f65965d1ebb37d05a83c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=x(x^3-2x^2-13x-10)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"240\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora que pegamos o fator comum de x, calculamos as ra\u00edzes ou zeros do polin\u00f4mio entre par\u00eanteses usando o m\u00e9todo de Ruffini: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/factorisation-des-polynomes-de-ruffini-exercices-resolus-pdf.jpg\" alt=\"fatora\u00e7\u00e3o de polin\u00f4mios de Ruffini exerc\u00edcios resolvidos pdf\" class=\"wp-image-1328\" width=\"224\" height=\"286\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Assim as ra\u00edzes do polin\u00f4mio s\u00e3o aquelas que encontramos pelo m\u00e9todo de Ruffini mais x=0 do fator comum:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6fd9d39f84a3308b2a958a5b213fad9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=0 \\qquad x=-1 \\qquad x=-2 \\qquad x=5\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"304\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E finalmente, decompondo o polin\u00f4mio em fatores obtemos a seguinte express\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-03584cf729dad52d2ade9eacd54f47d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)= 1 \\cdot x \\cdot (x+1)\\cdot (x+2)\\cdot (x-5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"294\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-55af6b96ce9ae5be4c9bbc749586ac30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{P(x)= x(x+1)(x+2)(x-5)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"234\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 4<\/h3>\n<p> Transforme o seguinte polin\u00f4mio de terceiro grau em fatores: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2eb940210fd5725256a4886885d4f990_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=6x^3+25x^2+21x-10\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"234\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__DDF5FF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#DDF5FF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Este polin\u00f4mio possui um termo independente, portanto calculamos suas ra\u00edzes com o algoritmo de Ruffini: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/factoriser-des-polynomes-de-degre-3-en-ligne-2.jpg\" alt=\"polin\u00f4mios fatoriais de grau 3 na linha 2\" class=\"wp-image-1334\" width=\"219\" height=\"124\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Por\u00e9m, quando chegamos a este ponto, n\u00e3o podemos continuar a aplicar a regra de Ruffini, pois sem outro n\u00famero inteiro o resto da divis\u00e3o \u00e9 zero.<\/p>\n<p class=\"has-text-align-left\"> Portanto, definimos o polin\u00f4mio resultante igual a zero:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b54e68406556ca63598a2f81ebd4bcac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6x^2+13x-5=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"139\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E aplicamos a f\u00f3rmula das equa\u00e7\u00f5es quadr\u00e1ticas para resolver a equa\u00e7\u00e3o resultante: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-982cd2e82d8511ceb1f93648c3ee61df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= \\cfrac{-b \\pm \\sqrt{b^2-4ac}}{2a}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"165\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-432e41dd639800e6ae78911e6bdbd440_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= \\cfrac{-13 \\pm \\sqrt{13^2-4\\cdot 6\\cdot (-5)}}{2\\cdot 6}= \\cfrac{-13\\pm \\sqrt{169+120}}{12} = \\cfrac{-13 \\pm\\sqrt{289}}{12}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"546\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-77785f81018b1d7a46a83d1567af638e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle x = \\cfrac{-13 \\pm 17}{12} = \\begin{cases}  \\cfrac{-13+17}{12} = \\cfrac{4}{12} = \\cfrac{1}{3} \\\\[4ex]\\cfrac{-13-17}{12} = \\cfrac{-30}{12} = -\\cfrac{5}{2} \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"348\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> As ra\u00edzes ou zeros do polin\u00f4mio s\u00e3o, portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-208bbe3d807db642e7f3cf8f0245c014_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=-2 \\qquad x=\\cfrac{1}{3} \\qquad x=-\\cfrac{5}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"230\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto a fatora\u00e7\u00e3o do polin\u00f4mio deve ser feita com fra\u00e7\u00f5es:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0a0cf1c1c64a18689df2941011dd3389_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle P(x)= 6\\left(x+2\\right)\\left(x-\\frac{1}{3}\\right)\\left(x+\\frac{5}{2}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"272\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 5<\/h3>\n<p> Determine a fatora\u00e7\u00e3o do seguinte polin\u00f4mio de grau 6: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aff4a9e14858bdbfee31195fdfa05b1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=x^6-3x^5+14x^3-12x^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"241\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__DDF5FF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#DDF5FF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> O polin\u00f4mio do problema n\u00e3o possui termo independente, ent\u00e3o devemos primeiro extrair o fator comum, que neste caso \u00e9 <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4c96aa574fedc2cca3206c8aacdd0255_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2:\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"27\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a4f53c4ed9cd82f1fcd39ecef4480bac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=x^2(x^4-3x^3+14x-12)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"247\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E uma vez removido o fator comum do polin\u00f4mio, encontramos as ra\u00edzes do polin\u00f4mio entre par\u00eanteses usando a regra de Ruffini: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/factoriser-des-polynomes-uniques.jpg\" alt=\"fatora\u00e7\u00e3o de polin\u00f4mios \u00fanicos\" class=\"wp-image-1337\" width=\"251\" height=\"206\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Mas quando chegamos a esta fase, n\u00e3o podemos continuar a avan\u00e7ar, porque sem outro n\u00famero inteiro, o resto \u00e9 zero.<\/p>\n<p class=\"has-text-align-left\"> Portanto, definimos o polin\u00f4mio obtido igual a zero:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-592ad3827d86125b15a72e4c8e8c5ac8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-4x+6=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"122\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E resolvemos a equa\u00e7\u00e3o quadr\u00e1tica com a f\u00f3rmula: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-982cd2e82d8511ceb1f93648c3ee61df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= \\cfrac{-b \\pm \\sqrt{b^2-4ac}}{2a}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"165\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e7cec78dac4f876f315c815297bbb0ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= \\cfrac{-(-4) \\pm \\sqrt{(-4)^2-4\\cdot 1\\cdot 6}}{2\\cdot 1}= \\cfrac{4\\pm \\sqrt{16-24}}{2} = \\cfrac{4 \\pm\\sqrt{-8}}{2} \\ \\color{red} \\bm{\\times}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"517\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> N\u00e3o existem ra\u00edzes de n\u00fameros negativos, portanto a equa\u00e7\u00e3o n\u00e3o tem solu\u00e7\u00e3o, o que significa que n\u00e3o podemos encontrar mais ra\u00edzes do polin\u00f4mio. Em outras palavras, o polin\u00f4mio n\u00e3o \u00e9 completamente fator\u00e1vel.<\/p>\n<p class=\"has-text-align-left\"> No entanto, as ra\u00edzes que conseguimos encontrar s\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1373d97d08d53d49fdf8b4f227f5d656_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=0 \\qquad x=0 \\qquad x=1 \\qquad x=-2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"290\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Observe que a raiz<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 repetido duas vezes porque removemos o fator comum de<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-09f6edd3d7af07ab26b4a0a71c20c0b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2,\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"22\" style=\"vertical-align: -4px;\"><\/p>\n<p> e como \u00e9 elevado ao quadrado, isso implica que \u00e9 uma raiz dupla.<\/p>\n<p class=\"has-text-align-left\"> Concluindo, o polin\u00f4mio fatorado ser\u00e1 o produto de todas as ra\u00edzes encontradas expressas como fatores<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f629cb3501652d3b8e4d6a30d92b5d4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x-a)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"53\" style=\"vertical-align: -5px;\"><\/p>\n<p> multiplicado pelo polin\u00f4mio obtido da regra de Ruffini que n\u00e3o p\u00f4de ser levado em considera\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d97b2a369799813209a1ba3e168ed71f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)= 1 \\cdot x^2 \\cdot (x-1)\\cdot (x+2)\\cdot (x^2-4x+6)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"350\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7738a261a919bbff02a7ed19dd408f28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{P(x)= x^2(x-1)(x+2)(x^2-4x+6)}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"290\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 6<\/h3>\n<p> Execute as fatora\u00e7\u00f5es de todos os seguintes polin\u00f4mios: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eb4c5b02f2cac45cb04ff218fb33a38d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ P(x)=x^2 + 12x+36\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"195\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5266319b8b4b96bf82cba4cabc3c4830_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ Q(x)=x^2 -64\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"144\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aa3aa86769d7a654e72a894bccb94fa1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ R(x)=x^2 - 18x+81\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"193\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f80148e5fc14408e7ec33fa35f594bdb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ S(x)=x^2+10x+24\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"193\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__DDF5FF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#DDF5FF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> O polin\u00f4mio na se\u00e7\u00e3o A) corresponde a uma identidade not\u00e1vel, notadamente o quadrado da soma. Sua fatora\u00e7\u00e3o \u00e9 portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0540de8a5f532fb36658af4c9af59dca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=x^2 + 12x+36 = \\bm{(x+6)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"253\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O polin\u00f4mio da se\u00e7\u00e3o B) tamb\u00e9m \u00e9 um produto not\u00e1vel, em particular \u00e9 a diferen\u00e7a dos quadrados, portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4903083e3b92847cafe379abb0c816a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Q(x)=x^2 -64 = \\bm{(x+8)(x-8)}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"251\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Da mesma forma, o polin\u00f4mio na se\u00e7\u00e3o C) \u00e9 uma igualdade not\u00e1vel, em particular consiste no quadrado de uma subtra\u00e7\u00e3o. Sua fatora\u00e7\u00e3o \u00e9 portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ce680ba71785064da5c773cda5916be0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"R(x)=x^2 - 18x+81 = \\bm{(x-9)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"253\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Finalmente, o polin\u00f4mio na parte D) n\u00e3o \u00e9 uma identidade not\u00e1vel. Devemos, portanto, igualar o polin\u00f4mio a 0 e resolver a equa\u00e7\u00e3o resultante para encontrar suas ra\u00edzes:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-48364ae74194a23aa761f92c1d5ddc7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2+10x+24 =0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"139\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Usamos a f\u00f3rmula da equa\u00e7\u00e3o quadr\u00e1tica: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-982cd2e82d8511ceb1f93648c3ee61df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= \\cfrac{-b \\pm \\sqrt{b^2-4ac}}{2a}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"165\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e5c923f0dcbdc9b54568db04cea19263_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= \\cfrac{-10 \\pm \\sqrt{10^2-4\\cdot 1\\cdot 24}}{2\\cdot 1}= \\cfrac{-10\\pm \\sqrt{100-96}}{2} = \\cfrac{-10 \\pm\\sqrt{4}}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"498\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-88934cfbe80af987a03e4fb1a2a72aa7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle x = \\cfrac{-10 \\pm 2}{2} = \\begin{cases}  \\cfrac{-10+2}{2} = \\cfrac{-8}{2} = -4 \\\\[4ex]\\cfrac{-10-2}{2} = \\cfrac{-12}{2} = -6\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"328\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> As ra\u00edzes do polin\u00f4mio D) s\u00e3o portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef7500f97e6fbdb358f7a4e39d4f33df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=-4 \\qquad x=-6\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"149\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E finalmente, o resultado da fatora\u00e7\u00e3o polinomial \u00e9:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e4c31b7f7d37fa2927e3bc07a2ebb18f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{S(x)=(x+4)(x+6)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"168\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<div id=\"ezoic-pub-ad-placeholder-176\" data-inserter-version=\"-1\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina explicamos como fatorar qualquer tipo de polin\u00f4mio. Veremos primeiro como fatorar um polin\u00f4mio com a regra de Ruffini, depois passaremos a como s\u00e3o fatorados polin\u00f4mios sem termo independente, depois analisaremos as fatora\u00e7\u00f5es de polin\u00f4mios raiz com fra\u00e7\u00f5es e, por fim, os casos especiais de fatora\u00e7\u00f5es (not\u00e1veis identidades, fatora\u00e7\u00e3o por agrupamento, trin\u00f4mios, etc.). Todas &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/\"> <span class=\"screen-reader-text\">Como fatorar polin\u00f4mios (fatora\u00e7\u00e3o polinomial)<\/span> Leia mais &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[38],"tags":[],"class_list":["post-57","post","type-post","status-publish","format-standard","hentry","category-tipos-de-polinomios"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Como fatorar polin\u00f4mios (fatora\u00e7\u00e3o polinomial) -<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Como fatorar polin\u00f4mios (fatora\u00e7\u00e3o polinomial) -\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina explicamos como fatorar qualquer tipo de polin\u00f4mio. Veremos primeiro como fatorar um polin\u00f4mio com a regra de Ruffini, depois passaremos a como s\u00e3o fatorados polin\u00f4mios sem termo independente, depois analisaremos as fatora\u00e7\u00f5es de polin\u00f4mios raiz com fra\u00e7\u00f5es e, por fim, os casos especiais de fatora\u00e7\u00f5es (not\u00e1veis identidades, fatora\u00e7\u00e3o por agrupamento, trin\u00f4mios, etc.). Todas &hellip; Como fatorar polin\u00f4mios (fatora\u00e7\u00e3o polinomial) Leia mais &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-09-17T07:24:22+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/factorisation-de-polynomes-factorisation.png\" \/>\n<meta name=\"author\" content=\"Equipe Mathoridade\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"Equipe Mathoridade\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. tempo de leitura\" \/>\n\t<meta name=\"twitter:data2\" content=\"16 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/\"},\"author\":{\"name\":\"Equipe Mathoridade\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\"},\"headline\":\"Como fatorar polin\u00f4mios (fatora\u00e7\u00e3o polinomial)\",\"datePublished\":\"2023-09-17T07:24:22+00:00\",\"dateModified\":\"2023-09-17T07:24:22+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/\"},\"wordCount\":3250,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"articleSection\":[\"Tipos de polin\u00f4mios\"],\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/\",\"url\":\"https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/\",\"name\":\"Como fatorar polin\u00f4mios (fatora\u00e7\u00e3o polinomial) -\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/#website\"},\"datePublished\":\"2023-09-17T07:24:22+00:00\",\"dateModified\":\"2023-09-17T07:24:22+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/#breadcrumb\"},\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathority.org\/pt\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Como fatorar polin\u00f4mios (fatora\u00e7\u00e3o polinomial)\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathority.org\/pt\/#website\",\"url\":\"https:\/\/mathority.org\/pt\/\",\"name\":\"Mathority\",\"description\":\"Onde a curiosidade encontra o c\u00e1lculo!\",\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathority.org\/pt\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"pt-BR\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathority.org\/pt\/#organization\",\"name\":\"Mathority\",\"url\":\"https:\/\/mathority.org\/pt\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pt-BR\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png\",\"contentUrl\":\"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png\",\"width\":703,\"height\":151,\"caption\":\"Mathority\"},\"image\":{\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/\"}},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\",\"name\":\"Equipe Mathoridade\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pt-BR\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"caption\":\"Equipe Mathoridade\"},\"sameAs\":[\"http:\/\/mathority.org\/pt\"]}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Como fatorar polin\u00f4mios (fatora\u00e7\u00e3o polinomial) -","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/","og_locale":"pt_BR","og_type":"article","og_title":"Como fatorar polin\u00f4mios (fatora\u00e7\u00e3o polinomial) -","og_description":"Nesta p\u00e1gina explicamos como fatorar qualquer tipo de polin\u00f4mio. Veremos primeiro como fatorar um polin\u00f4mio com a regra de Ruffini, depois passaremos a como s\u00e3o fatorados polin\u00f4mios sem termo independente, depois analisaremos as fatora\u00e7\u00f5es de polin\u00f4mios raiz com fra\u00e7\u00f5es e, por fim, os casos especiais de fatora\u00e7\u00f5es (not\u00e1veis identidades, fatora\u00e7\u00e3o por agrupamento, trin\u00f4mios, etc.). Todas &hellip; Como fatorar polin\u00f4mios (fatora\u00e7\u00e3o polinomial) Leia mais &raquo;","og_url":"https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/","article_published_time":"2023-09-17T07:24:22+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/factorisation-de-polynomes-factorisation.png"}],"author":"Equipe Mathoridade","twitter_card":"summary_large_image","twitter_misc":{"Escrito por":"Equipe Mathoridade","Est. tempo de leitura":"16 minutos"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/"},"author":{"name":"Equipe Mathoridade","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00"},"headline":"Como fatorar polin\u00f4mios (fatora\u00e7\u00e3o polinomial)","datePublished":"2023-09-17T07:24:22+00:00","dateModified":"2023-09-17T07:24:22+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/"},"wordCount":3250,"commentCount":0,"publisher":{"@id":"https:\/\/mathority.org\/pt\/#organization"},"articleSection":["Tipos de polin\u00f4mios"],"inLanguage":"pt-BR","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/","url":"https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/","name":"Como fatorar polin\u00f4mios (fatora\u00e7\u00e3o polinomial) -","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/#website"},"datePublished":"2023-09-17T07:24:22+00:00","dateModified":"2023-09-17T07:24:22+00:00","breadcrumb":{"@id":"https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/#breadcrumb"},"inLanguage":"pt-BR","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathority.org\/pt\/exemplos-de-fatoracao-de-polinomios-e-fatoracao-de-exercicios-resolvidos\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathority.org\/pt\/"},{"@type":"ListItem","position":2,"name":"Como fatorar polin\u00f4mios (fatora\u00e7\u00e3o polinomial)"}]},{"@type":"WebSite","@id":"https:\/\/mathority.org\/pt\/#website","url":"https:\/\/mathority.org\/pt\/","name":"Mathority","description":"Onde a curiosidade encontra o c\u00e1lculo!","publisher":{"@id":"https:\/\/mathority.org\/pt\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathority.org\/pt\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"pt-BR"},{"@type":"Organization","@id":"https:\/\/mathority.org\/pt\/#organization","name":"Mathority","url":"https:\/\/mathority.org\/pt\/","logo":{"@type":"ImageObject","inLanguage":"pt-BR","@id":"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/","url":"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png","contentUrl":"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png","width":703,"height":151,"caption":"Mathority"},"image":{"@id":"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/"}},{"@type":"Person","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00","name":"Equipe Mathoridade","image":{"@type":"ImageObject","inLanguage":"pt-BR","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","caption":"Equipe Mathoridade"},"sameAs":["http:\/\/mathority.org\/pt"]}]}},"yoast_meta":{"yoast_wpseo_title":"","yoast_wpseo_metadesc":"","yoast_wpseo_canonical":""},"_links":{"self":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts\/57","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/comments?post=57"}],"version-history":[{"count":0,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts\/57\/revisions"}],"wp:attachment":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/media?parent=57"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/categories?post=57"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/tags?post=57"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}